Problem 2. In this problem, we will practice a bit with numerical integration. We take the following facts for granted: for any a
Problem 2. In this problem, we will practice a bit with numerical integration. We take the following facts for granted: for any a
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 44RE
Related questions
Question
![Problem 2. In this problem, we will practice a bit with numerical integration. We take
the following facts for granted: for any a <b, any f: [a, b] → R, and any n ≥ 1,
and
RIGHT(ƒ,n) – [*° f(x)dx ≤
2
1(b-a)²
n
Compute
SIMP(f,n) - [ f(x)dx ≤ max f(i)(x)\,
[ab]
max f'(x)|
z€[a,b]
(b-a)5
180n4
SIMP(f,n):
(S)
where RIGHT(f, n) is the right hand Riemann sum with n equal intervals and SIMP (f, n)
is the approximation of f f(x)dx using Simpson's rule with n equal intervals. Here f(iv)
is the fourth derivative of f.
(i) Suppose that f is a third degree polynomial f(x) = a3x³ + a₂x² + a₁ + ao. Argue
that, for any n, a, and b,
(L)
f(x)dr.
(ii) Show that the same is not true for RIGHT(f, n) by giving an example of a third degree
polynomial f, a natural number n, and endpoints a < b so that
RIGHT (f, n) + f(z)dr.
Hint: linear functions are third degree polynomials and n = 1 is a natural number...
(iii) For the remaining parts, use a = 0, b = 2, and
f(x)=
-4.
RIGHT(f, 4)
and
SIMP (f, 4).
You may do this by hand, by writing code, or by using an online calculator, but you
must document your computations (either via screenshots or by writing down your
work).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b0d42b5-3479-420b-a2e4-bea672ad5500%2F78c5d8d1-87da-439f-b39a-04b3db72e359%2F6riz8pa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 2. In this problem, we will practice a bit with numerical integration. We take
the following facts for granted: for any a <b, any f: [a, b] → R, and any n ≥ 1,
and
RIGHT(ƒ,n) – [*° f(x)dx ≤
2
1(b-a)²
n
Compute
SIMP(f,n) - [ f(x)dx ≤ max f(i)(x)\,
[ab]
max f'(x)|
z€[a,b]
(b-a)5
180n4
SIMP(f,n):
(S)
where RIGHT(f, n) is the right hand Riemann sum with n equal intervals and SIMP (f, n)
is the approximation of f f(x)dx using Simpson's rule with n equal intervals. Here f(iv)
is the fourth derivative of f.
(i) Suppose that f is a third degree polynomial f(x) = a3x³ + a₂x² + a₁ + ao. Argue
that, for any n, a, and b,
(L)
f(x)dr.
(ii) Show that the same is not true for RIGHT(f, n) by giving an example of a third degree
polynomial f, a natural number n, and endpoints a < b so that
RIGHT (f, n) + f(z)dr.
Hint: linear functions are third degree polynomials and n = 1 is a natural number...
(iii) For the remaining parts, use a = 0, b = 2, and
f(x)=
-4.
RIGHT(f, 4)
and
SIMP (f, 4).
You may do this by hand, by writing code, or by using an online calculator, but you
must document your computations (either via screenshots or by writing down your
work).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Functions and Change: A Modeling Approach to Coll…
Algebra
ISBN:
9781337111348
Author:
Bruce Crauder, Benny Evans, Alan Noell
Publisher:
Cengage Learning
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Functions and Change: A Modeling Approach to Coll…
Algebra
ISBN:
9781337111348
Author:
Bruce Crauder, Benny Evans, Alan Noell
Publisher:
Cengage Learning
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
